JBMO 2017 Problem 3

Let $A B C$ be an acute triangle such that $A B \neq A C$, with circumcircle $\Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $B C$ and $D$ be a point on $\Gamma$ such that $A D \perp B C$. Let $T$ be a point such that $B D C T$ is a parallelogram and $Q$ a point on the same side of $B C$ as $A$ such that

$$\angle B Q M=\angle B C A \text { and } \angle C Q M=\angle C B A$$

Let the line $A O$ intersect $\Gamma$ at $E,(E \neq A)$ and let the circumcircle of $\triangle E T Q$ intersect $\Gamma$ at point $X \neq E$. Prove that the points $A, M$, and $X$ are collinear.